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Finding Parameters of Axis Aligned Ellipse from Implicit Coordinate Equation

  1. As given in Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses, the Implicit Coordinate Equation for Axis Aligned Ellipses are given as

    Ax2+Cy2+Dx+Ey+F=0   ...(1)

    Ax2+Dx+Cy2+Ey+F=0

    A(x2+DAx)+C(y2+ECy)+F=0   ...(2)

    Completing the Squares in equation (2) we get

    A(x2+2D2Ax+D24A2D24A2)+C(y2+2E2Cy+E24C2E24C2)+F=0

    A(x2+2D2Ax+D24A2)+C(y2+2E2Cy+E24C2)+FD24AE24C=0

    A(x+D2A)2+C(y+E2C)2+FD24AE24C=0

    A(x+D2A)2+C(y+E2C)2=D24A+E24CF   ...(3)

    Setting D24A+E24CF=G in equation (3) we get

    A(x+D2A)2+C(y+E2C)2=G

    A(x+D2A)2G+C(y+E2C)2G=1

    (x+D2A)2GA+(y+E2C)2GC=1   ...(4)

    If equation (1) represents an Implicit Equation of Ellipse (Real or Imaginary), values of GA and GC given in equation (4) will have Same Sign.

    If the values of GA>0 and GC>0, then the equation (1) represents a Real Ellipse and equation (4) gives it's Stardard Coordinate Equation.

    If the values of GA<0 and GC<0, then the equation (1) represents an Imaginary Ellipse and equation (4) gives it's Stardard Coordinate Equation.
  2. Using equation (4) Various Parameters of Ellipse given by Implicit Equation (1) can be given as
    1. Coordinates of the Center: (xc,yc)=(D2A,E2C)
    2. Length of Semi-Major Axis: a=max(|GA|,|GC|)
    3. Length of Major Axis: L=2a
    4. Length of Semi-Minor Axis: b=min(|GA|,|GC|)
    5. Length of Minor Axis: l=2b
    6. Distance Between 2 Foci: F=L2l2=2c=2a2b2
    7. Eccentricity e : FL=ca
    8. Length for Latus Rectum : l2L=2b2a
    9. If Major Axis is Parallel to X-Axis then
      Angular Orientation: 0°
      Coordinates of the 2 Foci: (xc+c,yc),(xcc,yc)
      Coordinates of the 2 Vertices: (xc+a,yc),(xca,yc)
      Coordinates of the 2 Co-Vertices: (xc,yc+b),(xc,ycb)
      Equation of Major Axis: y=yc
      Equation of Minor Axis: x=xc
      Equation of 2 Directrices: x=xc+L22F,x=xcL22F   OR   x=xc+a2c,x=xca2c
      Equation of 2 Latus Recta: x=xc+c,x=xcc
      Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: (xc+c,yc+l22L),(xc+c,ycl22L)   OR   (xc+c,yc+b2a),(xc+c,ycb2a)
      Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: (xcc,yc+l22L),(xcc,ycl22L)   OR   (xcc,yc+b2a),(xcc,ycb2a)
    10. If Major Axis is Parallel to Y-Axis then
      Angular Orientation: 90°
      Coordinates of the 2 Foci: (xc,yc+c),(xc,ycc)
      Coordinates of the 2 Vertices: (xc,yc+a),(xc,yca)
      Coordinates of the 2 Co-Vertices: (xc+b,yc),(xcb,yc)
      Equation of Major Axis: x=xc
      Equation of Minor Axis: y=yc
      Equation of 2 Directrices: y=yc+L22F,y=ycL22F   OR   y=yc+a2c,y=yca2c
      Equation of 2 Latus Recta: y=yc+c,y=ycc
      Coordinates of Points of Intersection of Latus Rectum-1 and Ellipse: (xc+l22L,yc+c),(xcl22L,yc+c)   OR   (xc+b2a,yc+c),(xcb2a,yc+c)
      Coordinates of Points of Intersection of Latus Rectum-2 and Ellipse: (xc+l22L,ycc),(xcl22L,ycc)   OR   (xc+b2a,ycc),(xcb2a,ycc)
Related Topics
Derivation of Standard and Implicit Coordinate Equation for Axis Aligned Ellipses,    Introduction to Ellipse and Imaginary Ellipse,    General Quadratic Equations in 2 Variables and Conic Sections
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